Simplify the following formula with Gamma functions

In summary, the solution is that the middle expression is the same as the first, but with the Gamma function included.
  • #1
montoyas7940
364
21
[tex]\frac{\beta^\alpha \Gamma(\alpha + 1)}{\Gamma (\alpha) \beta^{\alpha+ 1}}[/tex]

= [tex]\frac{\alpha \Gamma (\alpha)}{ \beta \Gamma (\alpha)}[/tex]

= [tex]\frac{\alpha}{ \beta}[/tex]

This is the solution. In trying to get the middle expression out of the first I quickly end up with a mess. How should I approach this?
 
Last edited:
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  • #2
Please try to use LaTeX tags, or write the fractions on one line using brackets. So either write
[tex]\frac{B^a C(a + 1)}{C(a) B^(a + 1)}[/tex]
or write
B^a C(a + 1) / [ C(a) B^(a+1) ]

Also, what is C(a)? Is it C multiplied by a? Or is C a function and is C(a) the function value in a? Or did you forget a caret and did you mean C^a ? In the last case I get C/B which is closest to the supposed solution you gave.
 
  • #3
Thanks, CompuChip.
As it turns out there is more to this that I was not given. [tex]\Gamma(\alpha)[/tex] is a function. [tex]\Gamma(n)[/tex] = (n-1)! (factorial).
 
  • #4
Yes, yes, the lovely Gamma function. Is this for a statistics class?
Did you solve it after you found out about the Gamma function?
CC
 
  • #5
Probability for risk management, Happyg1. Not a class though, just for kicks.
I haven't solved it yet, I had to walk away for a while...
 
  • #7
We just got finished studying the Gamma Distribution in Mathematical Stats class. It's an interesting distribution. We went through the entire derivation of the properties that tiny-tim is showing you. Very cool...and kinda morbid. Our Professor explained that the Gamma distribution is the distribution used for "life testing"...the waiting time until death. Now you know about a function that models the waiting time until a "success" (which is death) occurs.
I love math!
CC
 
  • #8
So,

[tex]\frac{\beta^\alpha \alpha\Gamma(\alpha)}{\beta^{\alpha+ 1}\Gamma (\alpha) }[/tex]

= [tex]\frac{\alpha \Gamma (\alpha)}{ \beta \Gamma (\alpha)}[/tex]

= [tex]\frac{\alpha}{ \beta}[/tex]

I like it! Maybe I will "survive".
 
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1. What are Gamma functions?

Gamma functions are mathematical functions that are used to extend the concept of factorial to non-integer values. They are denoted by the Greek letter gamma (γ) and are commonly used in various areas of mathematics, physics, and engineering.

2. How do Gamma functions simplify formulas?

Gamma functions can be used to simplify formulas by replacing complex expressions involving factorials with simpler expressions involving gamma functions. This is especially useful when working with non-integer values.

3. Can all formulas be simplified with Gamma functions?

No, not all formulas can be simplified with Gamma functions. These functions are most commonly used to simplify expressions involving factorials or other special functions. They may not be applicable to all types of formulas.

4. Are there any limitations to using Gamma functions?

Yes, there are some limitations to using Gamma functions. One limitation is that they are only defined for positive real numbers, so they cannot be used for negative or complex values. Additionally, some formulas may not have a simplified expression using Gamma functions.

5. Can Gamma functions be used in practical applications?

Yes, Gamma functions have many practical applications in fields such as physics, engineering, and statistics. They are commonly used to solve problems involving probability, distributions, and complex integrals. They are also used in the development of mathematical models and algorithms.

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